Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | #432 |
| prime radical | C25 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,19,14,20,29,37,15,17,28,34,12,18,26,35)(11,21,24,38,25,16,30,32,22,33,27,40,23,31,39,36);(2,4)(9,11)(10,16,17,31)(12,23)(13,27)(14,25)(15,22)(18,21,20,33)(19,36,34,32)(24,28)(26,30)(29,39)(35,40,37,38);(6,8)(10,17)(11,22)(13,28)(18,20)(23,25)(26,29)(32,36)(38,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,g^a=gi,c^b=ci,f^b=fi,f^c=fi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53577);